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The-number-of-solutions-of-equations-13-18tan-x-6tan-x-3-where-2pi-lt-x-lt-2pi-is-




Question Number 138999 by bramlexs22 last updated on 21/Apr/21
The number of solutions  of equations (√(13−18tan x)) = 6tan x−3  where −2π<x<2π is
$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions} \\ $$$$\mathrm{of}\:\mathrm{equations}\:\sqrt{\mathrm{13}−\mathrm{18tan}\:\mathrm{x}}\:=\:\mathrm{6tan}\:\mathrm{x}−\mathrm{3} \\ $$$$\mathrm{where}\:−\mathrm{2}\pi<\mathrm{x}<\mathrm{2}\pi\:\mathrm{is} \\ $$
Answered by EDWIN88 last updated on 21/Apr/21
(1) 13−18tan x≥0 ; tan x≤((13)/(18))  (2) 13−18tan x=36tan^2 x−36tan x+9  36tan^2 x−18tan x−4=0  18tan^2 x−9tan x−2=0  tan x = ((9 ± 15)/(36)) = { ((tan x=(2/3))),((tan x=−(1/6))) :}
$$\left(\mathrm{1}\right)\:\mathrm{13}−\mathrm{18tan}\:\mathrm{x}\geqslant\mathrm{0}\:;\:\mathrm{tan}\:\mathrm{x}\leqslant\frac{\mathrm{13}}{\mathrm{18}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{13}−\mathrm{18tan}\:\mathrm{x}=\mathrm{36tan}\:^{\mathrm{2}} \mathrm{x}−\mathrm{36tan}\:\mathrm{x}+\mathrm{9} \\ $$$$\mathrm{36tan}\:^{\mathrm{2}} \mathrm{x}−\mathrm{18tan}\:\mathrm{x}−\mathrm{4}=\mathrm{0} \\ $$$$\mathrm{18tan}\:^{\mathrm{2}} \mathrm{x}−\mathrm{9tan}\:\mathrm{x}−\mathrm{2}=\mathrm{0} \\ $$$$\mathrm{tan}\:\mathrm{x}\:=\:\frac{\mathrm{9}\:\pm\:\mathrm{15}}{\mathrm{36}}\:=\begin{cases}{\mathrm{tan}\:\mathrm{x}=\frac{\mathrm{2}}{\mathrm{3}}}\\{\mathrm{tan}\:\mathrm{x}=−\frac{\mathrm{1}}{\mathrm{6}}}\end{cases} \\ $$

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